Question

In Exercises \(16-19\) find all values of \(\omega\) such that boundary problem has a unique solution, and find the solution by the method used to prove Theorem 13.1.3. For other values of \(\omega\), find conditions on \(F\) such that the problem has a solution, and find all solutions by the method used to prove Theorem \(13.1 .4 .\) $$ y^{\prime \prime}+\omega^{2} y=F(x), \quad y^{\prime}(0)=0, \quad y(\pi)=0 $$

Short answer

Solution: The given boundary value problem has a unique solution for values of \(\omega\) such that \(\omega \neq n\), for all \(n \in \mathbb{Z}\). For \(\omega = n, n \in \mathbb{Z}\), the problem has a solution if the forcing function \(F(x)\) satisfies the condition $$ \int\left[ \int F(x) \right] dx\Big|_{\pi} = A\cos(n\pi). $$

Step-by-step solution

Determine the homogeneous ODE and solutions

The given ODE is: $$ y^{\prime \prime}+\omega^{2} y=F(x) $$ First, let us consider the corresponding homogeneous equation: $$ y^{\prime \prime}+\omega^{2} y=0 $$ The general solutions of this ODE are: $$ y_{hom}(x) = A\cos(\omega x) + B\sin(\omega x) $$ where \(A\) and \(B\) are constants.

Apply boundary conditions to homogeneous equation

We are given the boundary conditions: $$ y^{\prime}(0)=0, \quad y(\pi)=0 $$ Applying these boundary conditions to the homogeneous solution, we have: $$ y^{\prime}(0)= -A\omega\sin(\omega*0) + B\omega\cos(\omega*0) = 0 $$ This implies that \(B=0\). Now, we can find \(A\) using the second boundary condition: $$ y(\pi) = A\cos(\omega \pi) = 0 $$

Determine the values of \(\omega\) for a unique solution

The equation \(A\cos(\omega\pi) = 0\) has a unique solution if \(\cos(\omega\pi) \neq 0\), which means \(\omega\pi \neq n\pi\) for all integers \(n\). Therefore, the values of \(\omega\) such that the boundary value problem has a unique solution are: $$ \omega \neq n \textrm{, } \forall n \in \mathbb{Z} $$

Find the general solution and apply boundary conditions

For the given values of \(\omega\), we can find the general solution of the original ODE by establishing a relationship between \(F(x)\) and its derivatives. We have: $$ y^{\prime \prime} + \omega^2 y = F(x) $$ Integrating both sides twice with respect to \(x\), we get: $$ y(x) = \int\left[ \int F(x) \right] dx + A\cos(\omega x) + B\sin(\omega x) $$ Now, we apply the boundary conditions: $$ y^{\prime}(0)=0 \Rightarrow B=0 $$ and $$ y(\pi)=0 \Rightarrow A\cos(\omega \pi) + \int\left[ \int F(x) \right] dx\Big|_{\pi} =0 $$

Find conditions on \(F\) such that the problem has a solution

For \(\omega = n, n \in \mathbb{Z}\), the problem has a solution if \(F(x)\) satisfies: $$ \int\left[ \int F(x) \right] dx\Big|_{\pi} = A\cos(n\pi) $$ This is the condition on \(F\) for which the problem has a solution.

Find all solutions by the method used to prove Theorem 13.1.4

Since \(A\cos(\omega \pi) = \int\left[ \int F(x) \right] dx\Big|_{\pi}\), we can rewrite the general solution as: $$ y(x) = \int\left[ \int F(x) \right] dx + A\cos(\omega x) $$ For the given values of \(\omega\), this expression gives all the solutions to the boundary value problem.

Key concepts

Homogeneous Differential Equations

In mathematics, homogeneous differential equations are pivotal in understanding a wide variety of problems dealing with rate of change. A differential equation is a mathematical equation that relates some function with its derivatives. In the context of the exercise under review, the equation: \[ y^{\prime \prime} + \omega^{2} y = F(x) \] is first considered in its homogeneous form, that is when \( F(x) = 0 \). This simplifies our equation to: \[ y^{\prime \prime} + \omega^{2} y = 0 \] This type of equation allows us to find the complementary function or solutions that are generated by the equation without any external force (or non-homogeneous term \(F(x)\)). For the given homogeneous differential equation, the general solution can be expressed as: \[ y_{hom}(x) = A\cos(\omega x) + B\sin(\omega x) \] where \( A \) and \( B \) are arbitrary constants determined by applying initial or boundary conditions. This solution represents all possible outcomes for varying constants, providing a fundamental building block toward solving more complex differential equations.

Boundary Conditions

Boundary conditions are essential in solving differential equations because they allow us to find specific solutions by determining the constants in the general solution. In the given exercise, we encounter two boundary conditions:

• \( y^{\prime}(0)=0 \)
• \( y(\pi)=0 \)

The boundary condition \( y^{\prime}(0)=0 \) leads to \( B = 0 \), simplifying our solution to just: \[ y_{hom}(x) = A\cos(\omega x) \] Furthermore, applying the second boundary condition \( y(\pi)=0 \) gives the equation \( A\cos(\omega \pi) = 0 \). Through these boundary conditions, we narrow down our infinite general solution to a specific case, which greatly simplifies solving the problem. Boundary conditions are a powerful tool in converting a general solution to a unique solution by eliminating any ambiguity regarding the constants in our solution.

Unique Solution

In mathematics, a unique solution of a boundary value problem ensures that for the given inputs there is exactly one solution based on the provided constraints. For the boundary value problem described in the exercise: \[ y^{\prime \prime} + \omega^{2} y = F(x) \] with conditions \( y^{\prime}(0)=0 \) and \( y(\pi)=0 \), the quest is to find all values of \( \omega \) for which there is a unique solution. A unique solution occurs when \( \cos(\omega \pi) eq 0 \), meaning \( \omega eq n \) where \( n \) is any integer. This restriction ensures that our previously derived formula \( A\cos(\omega \pi) = 0 \) holds non-trivially, with \( A eq 0 \). This criterion is crucial to differentiate between cases where multiple solutions exist versus a unique, well-defined outcome. Under these conditions, we apply the initial and boundary conditions to resolve the problem into a precise, coherent solution, eliminating superfluous or infinite solutions that would otherwise arise. By setting constraints for \( \omega \), we provide a clear and deterministic pathway to the unique solution of the differential equation.